MEI AS Further Pure Mathematics: Book 1 (MEI Structured by Catherine Berry

This sequence, popular for accessibility and for a student-friendly method, has a wealth of positive aspects: labored examples, actions, investigations, graded routines, Key issues summaries and dialogue issues. to make sure examination luck there are many updated examination query, plus indications to point universal pitfalls. MEI provide complete help to colleges via their community with newsletters, education days and an annual convention. extra natural arithmetic 1 is the middle part within the AS extra arithmetic qualification.

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29 The inverse of a product 1 Matrices ? ● How would you undo the effect of a rotation followed by a reflection? How would you write down the inverse of a matrix product MN in terms of M–1 and N–1? Suppose you want to find the inverse of the product AB, where A and B are non-singular matrices. This means that you need to find a matrix X such that X(AB) = I. X(AB) = I ⇒ XABB–1 = IB–1 ⇒ XA = B–1 ⇒ XAA–1 = B–1A–1 ⇒ X = B–1A–1 Post-multiply by B–1. Post-multiply by A–1. Thus (AB)–1 = B–1A–1, where A and B are non-singular matrices of the same order.

What do you notice? 20. This is called transforming the plane. 20 33 Each point on the image line P′Q′ is the image of infinitely many points. 20, and every point on AB maps to P′. Similarly, I′ is the image of all points on a line through I parallel to AB and each point on a line through Q parallel to AB maps to Q′. Matrices 1 ? ● What happens when you map any shape using the zero matrix ( 00 00 )? How is the plane transformed? 7 The plane is transformed by means of the matrix M = ( 42 –6–3 ).

Triangle U is obtained by rotating triangle S through 135° anticlockwise about the origin. (iii) Find the matrix which transforms triangle S into triangle U. (iv) Find the matrix which transforms triangle T into triangle U. [MEI, part] 6 The plane is transformed by means of the matrix M = (i) (ii) (iii) 7 (21 42 ). Show that det M = 0, and that the whole plane is mapped on to the line x – 2y = 0. The point P(x, y) is mapped to P′(4, 2). Use the equation p′ = Mp to show that P could be anywhere on the line x + 2y = 2.